Is it permissible to pay $180 to someone who promises to multiply the amount to more than $1000 after five months, claiming it is for charity, and would it be considered usury if the promised amount is recouped?

The transaction mentioned between your sister and this individual, whether it was a loan or a partnership, is impermissible. If it was a loan, it is an interest-based loan (ribawi loan) because it draws a benefit for the lender, and every loan that draws a benefit is usury (riba). If it was a partnership, it is invalid due to the guaranteed profit within it. If a partnership becomes invalid, the profits are distributed according to the capital of the partners, and the worker receives wages for his labor. Based on this, if it was an interest-based loan, your sister is entitled only to her principal. If it was a partnership with a guaranteed profit, it must be dissolved.